par Zhu, Jielin;Kuske, Rachel;Erneux, Thomas
Référence SIAM journal on applied dynamical systems, 14, 4, page (2030–2068)
Publication Publié, 2015-11-24
Article révisé par les pairs
Résumé : We consider the effect on tipping from an additive periodic forcing in a canonical model with asaddle node bifurcation and a slowly varying bifurcation parameter. Here tipping refers to thedramatic change in dynamical behavior characterized by a rapid transition away from a previouslyattracting state. In the absence of the periodic forcing, it is well known that a slowly varyingbifurcation parameter produces a delay in this transition, beyond the bifurcation point for thestatic case. Using a multiple scales analysis, we consider the effect of amplitude and frequency ofthe periodic forcing relative to the drifting rate of the slowly varying bifurcation parameter. Weshow that a high frequency oscillation drives an earlier tipping when the bifurcation parametervaries more slowly, with the advance of the tipping point proportional to the square of the ratio ofamplitude to frequency. In the low frequency case the position of the tipping point is affected bythe frequency, amplitude, and phase of the oscillation. The results are based on an analysis of thelocal concavity of the trajectory, used for low frequencies both of the same order as the drifting rateof the bifurcation parameter and for low frequencies larger than the drifting rate. The tipping pointlocation is advanced with increased amplitude of the periodic forcing, with critical amplitudes wherethere are jumps in the location, yielding significant advances in the tipping point. We demonstratethe analysis for two applications with saddle node–type bifurcations.